Trying to figure out the best way to start this blog was difficult, since I’ve been trying to pinpoint what audience I’m writing for–students, mathematicians, or the general public. I’ve begun to realize the answer is “all of the above”–I want to be able to elaborate on examples that I talk about in class; I want to be able to talk through the math I’m currently reading and my research; and I want to be able to have something to show my friends and family about what kinds of flavors of math exist beyond what they teach in schools. So while the level of background for the topics might change drastically from post to post, the idea is that there will be something here for everybody (and with the caveat that I can’t promise to post regularly, as a respectable blog would).
There are several places I could choose to begin: What’s my story? What got me into math? What am I up to right now? But instead I am going to start at the beginning of a common conversation I have with someone I’m meeting for the first time:
Them: Oh, you study math! What sort of math is it that you do?
This is very representative of the first few conversations I had like this. I didn’t know how to answer since the general population has usually hazy memories of functions, and perhaps knows some calculus. This blog post is what I developed over the course of many conversations as answers to this question, with the best stick figures I can muster.
Flavor 1 (Operator algebras)
Answer A: So you know how a line is 1-dimensional? Well there are a couple of things you can do with it: you can flip it (multiply by -1), you can stretch it out and in (also by multiplication). If you had a 2-dimensional plane, you might also be able to rotate it. All of these things go under the special name of linear transformations. I work with linear transformations, but on infinite dimensional spaces.
Answer B: (For those familiar with linear algebra) I do linear algebra on (possibly) infinite dimensional spaces.
Flavor 2 (Functional analysis)
Answer A: Functions are ways we can model different things in the world. For example, I could have a function that told me the population of bunnies on campus in a specific month, and it probably would be a curve that would go up and up over time. Some models can be more accurate than others, and you try to get functions that are closer and closer to what they should be. But notice, I just started talking about functions being close–so we have to have some notion of distance between functions. So we start thinking about functions being points in some space, and having a way to measure the distance between them. This new way of thinking about functions is an area I’m involved in.
Answer B: (For those familiar with series) The group of all power series that converge in a particular radius is an example of a group of functions that we can ask a lot of questions about as a group. We can also do things like add them together, and sometimes multiply, and using series we can also define a distance or norm. This can be a very powerful tool in studying functions we care about.
Flavor 3 (Noncommutative functions)
Answer A: Do you remember functions like x^2 or 3x+1 or xy-2? These are all commutative, because the variables are real/complex numbers and x times y is always the same as y times x. You can always switch the order of multiplication. But there are many examples of functions out there where you can’t switch the order of multiplication — these are called noncommutative. So I study these kinds of functions, try to develop calculus for them, try to do the same sorts of things we do with normal commutative ones.
An example of this is if you’re sending a signal through radio waves or something like that, you might try to send a big block of numbers to represent a picture or a bunch of data to later be decoded. In this case, these blocks of numbers are noncommutative. So, if you want to describe how the signal might change because of signal noise/interference, you need some sort of noncommutative calculus.
Answer B: (For those familar with matrices) …and an example of a noncommutative function is one with matrices for variables. When you have matrices, A times B is not usually the same thing as B times A, so things get messy.
Three is a nice number, so I’ll stop here for now.